Bloch constants of bounded symmetric domains

Author:
Genkai Zhang

Journal:
Trans. Amer. Math. Soc. **349** (1997), 2941-2949

MSC (1991):
Primary 32H02, 32M15

DOI:
https://doi.org/10.1090/S0002-9947-97-01518-3

MathSciNet review:
1329540

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $D_{1}$ and $D_{2}$ be two irreducible bounded symmetric domains in the complex spaces $V_{1}$ and $V_{2}$ respectively. Let $E$ be the Euclidean metric on $V_{2}$ and $h$ the Bergman metric on $V_{1}$. The Bloch constant $b(D_{1}, D_{2})$ is defined to be the supremum of $E(f^{\prime }(z)x, f^{\prime }(z)x)^{\frac {1}{2}}/h_{z}(x, x)^{1/2}$, taken over all the holomorphic functions $f: D_{1}\to D_{2}$ and $z\in D_{1}$, and nonzero vectors $x\in V_{1}$. We find the constants for all the irreducible bounded symmetric domains $D_{1}$ and $D_{2}$. As a special case we answer an open question of Cohen and Colonna.

- Joel M. Cohen and Flavia Colonna,
*Bounded holomorphic functions on bounded symmetric domains*, Trans. Amer. Math. Soc.**343**(1994), no. 1, 135–156. MR**1176085**, DOI https://doi.org/10.1090/S0002-9947-1994-1176085-6 - Sigurdur Helgason,
*Differential geometry, Lie groups, and symmetric spaces*, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR**514561** - L. K. Hua,
*Harmonic analysis of functions of several complex variables in the classical domains*, American Mathematical Society, Providence, R.I., 1963. Translated from the Russian by Leo Ebner and Adam Korányi. MR**0171936** - O. Loos,
*Bounded Symmetric Domains and Jordan Pairs*, University of California, Irvine, 1977. - Ottmar Loos,
*Jordan pairs*, Lecture Notes in Mathematics, Vol. 460, Springer-Verlag, Berlin-New York, 1975. MR**0444721** - Walter Rudin,
*Function theory in the unit ball of ${\bf C}^{n}$*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR**601594** - Ichirô Satake,
*Algebraic structures of symmetric domains*, Kanô Memorial Lectures, vol. 4, Iwanami Shoten, Tokyo; Princeton University Press, Princeton, N.J., 1980. MR**591460** - H. Upmeier,
*Jordan Algebras in Analysis, Operator Theory, and Quantum Mechanics*, Regional Conference Series in Mathematics No.67, Amer. Math. Soc., 1987. - Joseph A. Wolf,
*Fine structure of Hermitian symmetric spaces*, Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969–1970), Dekker, New York, 1972, pp. 271–357. Pure and App. Math., Vol. 8. MR**0404716** - Z. Yan,
*Extremal holomorphic mappings between a bounded symmetric domain and the unit ball*, preprint, Berkeley, 1993.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
32H02,
32M15

Retrieve articles in all journals with MSC (1991): 32H02, 32M15

Additional Information

**Genkai Zhang**

Affiliation:
School of Mathematics, University of New South Wales, Kensington NSW 2033, Australia

Address at time of publication:
Department of Mathematics, University of Karlstad, S-651 88 Karlstad, Sweden

Email:
genkai.zhang@hks.se

Keywords:
Bounded symmetric domain,
holomorphic mapping,
Schwarz lemma,
Bergman metric,
Bloch constant

Received by editor(s):
November 21, 1994

Received by editor(s) in revised form:
May 10, 1995

Additional Notes:
Research sponsored by the Australian Research Council

Article copyright:
© Copyright 1997
American Mathematical Society